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Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences
1. | Department of Politics, New York University, New York, NY 10012, USA |
2. | Department of Mathematics, Union College, Schenectady, NY 12308, USA |
3. | Faculties of Electrical Engineering and Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel |
4. | Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA |
5. | Department of Biostatistics, Brown University, Providence, RI 02912, USA |
We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ \epsilon\sim \rm{Bernoulli}\left(p\right) $ and $ p\in [0, 1] $ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.
References:
[1] |
G. Akemann, Z. Burda and M. Kieburg, Universal distribution of Lyapunov exponents for products of Ginibre matrices, J. Phys. A, 47 (2014), 395202, 35 pp.
doi: 10.1088/1751-8113/47/39/395202. |
[2] |
G. Akemann, M. Kieburg and L. Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A, 46 (2013), 275205, 22 pp.
doi: 10.1088/1751-8113/46/27/275205. |
[3] |
Y. Benoist and J.-F. Quint,
Central limit theorem for linear groups, Ann. Probab., 44 (2016), 1308-1340.
doi: 10.1214/15-AOP1002. |
[4] |
P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, 8. Birkhäuser Boston, Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9172-2. |
[5] |
P. Chassaing, G. Letac and M. Mora, Brocot sequences and random walks in SL(2, R), Probability Measures on Groups, VII (Oberwolfach, 1983), Lecture Notes in Math., Springer,
Berlin, 1064 (1984), 36–48.
doi: 10.1007/BFb0073632. |
[6] |
J. E. Cohen and C. M. Newman, The stability of large random matrices and their products, Ann. Probab., 12 (1984), 283–310, http://links.jstor.org.ezproxy.lib.uconn.edu/sici?sici=0091-1798(198405)12:2<283:TSOLRM>2.0.CO;2-O&origin=MSN.
doi: 10.1214/aop/1176993291. |
[7] |
P. J. Forrester,
Lyapunov exponents for products of complex Gaussian random matrices, J. Stat. Phys., 151 (2013), 796-808.
doi: 10.1007/s10955-013-0735-7. |
[8] |
P. J. Forrester, Asymptotics of finite system Lyapunov exponents for some random matrix ensembles, J. Phys. A, 48 (2015), 215205, 17 pp.
doi: 10.1088/1751-8113/48/21/215205. |
[9] |
H. Furstenberg and H. Kesten,
Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.
doi: 10.1214/aoms/1177705909. |
[10] |
H. Furstenberg and Y. Kifer,
Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.
doi: 10.1007/BF02760620. |
[11] |
A. Goswami, Random continued fractions: A Markov chain approach, Econom. Theory, 23 (2004), 85–105, https://doi.org/10.1007/BF02760620.
doi: 10.1007/s00199-002-0333-4. |
[12] |
É. Janvresse, B. Rittaud and T. de la Rue,
How do random Fibonacci sequences grow?, Probab. Theory Related Fields, 142 (2008), 619-648.
doi: 10.1007/s00440-007-0117-7. |
[13] |
É. Janvresse, B. Rittaud and T. de la Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, J. Phys. A, 42 (2009), 085005, 18 pp.
doi: 10.1088/1751-8113/42/8/085005. |
[14] |
É. Janvresse, B. Rittaud and T. de la Rue,
Almost-sure growth rate of generalized random Fibonacci sequences, Ann. Inst. Henri Poincaré Probab. Stat., 46 (2010), 135-158.
doi: 10.1214/09-AIHP312. |
[15] |
V. Kargin,
On the largest Lyapunov exponent for products of Gaussian matrices, J. Stat. Phys., 157 (2014), 70-83.
doi: 10.1007/s10955-014-1077-9. |
[16] |
M. Kieburg and H. Kösters,
Products of random matrices from polynomial ensembles, Ann. Inst. Henri Poincaré Probab. Stat., 55 (2019), 98-126.
doi: 10.1214/17-AIHP877. |
[17] |
R. Lima and M. Rahibe,
Exact Lyapunov exponent for infinite products of random matrices, J. Phys. A, 27 (1994), 3427-3437.
doi: 10.1088/0305-4470/27/10/019. |
[18] |
J. Marklof, Y. Tourigny and L. Wolowski,
Explicit invariant measures for products of random matrices, Trans. Amer. Math. Soc., 360 (2008), 3391-3427.
doi: 10.1090/S0002-9947-08-04316-X. |
[19] |
C. M. Newman, The distribution of Lyapunov exponents: Exact results for random matrices, Comm. Math. Phys., 103 (1986), 121–126, http://projecteuclid.org/euclid.cmp/1104114627.
doi: 10.1007/BF01464284. |
[20] |
Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, Lyapunov
Exponents (Oberwolfach, 1990), Lecture Notes in Math., Springer, Berlin, 1486 (1991), 64–
80.
doi: 10.1007/BFb0086658. |
[21] |
Y. Peres, Domains of analytic continuation for the top Lyapunov exponent, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 131–148, http://www.numdam.org/item?id=AIHPB_1992__28_1_131_0. |
[22] |
M. Pollicott,
Maximal Lyapunov exponents for random matrix products, Invent. Math., 181 (2010), 209-226.
doi: 10.1007/s00222-010-0246-y. |
[23] |
V. Yu. Protasov and R. M. Jungers,
Lower and upper bounds for the largest Lyapunov exponent of matrices, Linear Algebra Appl., 438 (2013), 4448-4468.
doi: 10.1016/j.laa.2013.01.027. |
[24] |
D. Ruelle,
Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80.
doi: 10.1016/0001-8708(79)90029-X. |
[25] |
D. Viswanath,
Random Fibonacci sequences and the number $1.13198824\ldots$, Math. Comp., 69 (2000), 1131-1155.
doi: 10.1090/S0025-5718-99-01145-X. |
show all references
References:
[1] |
G. Akemann, Z. Burda and M. Kieburg, Universal distribution of Lyapunov exponents for products of Ginibre matrices, J. Phys. A, 47 (2014), 395202, 35 pp.
doi: 10.1088/1751-8113/47/39/395202. |
[2] |
G. Akemann, M. Kieburg and L. Wei, Singular value correlation functions for products of Wishart random matrices, J. Phys. A, 46 (2013), 275205, 22 pp.
doi: 10.1088/1751-8113/46/27/275205. |
[3] |
Y. Benoist and J.-F. Quint,
Central limit theorem for linear groups, Ann. Probab., 44 (2016), 1308-1340.
doi: 10.1214/15-AOP1002. |
[4] |
P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Progress in Probability and Statistics, 8. Birkhäuser Boston, Inc., Boston, MA, 1985.
doi: 10.1007/978-1-4684-9172-2. |
[5] |
P. Chassaing, G. Letac and M. Mora, Brocot sequences and random walks in SL(2, R), Probability Measures on Groups, VII (Oberwolfach, 1983), Lecture Notes in Math., Springer,
Berlin, 1064 (1984), 36–48.
doi: 10.1007/BFb0073632. |
[6] |
J. E. Cohen and C. M. Newman, The stability of large random matrices and their products, Ann. Probab., 12 (1984), 283–310, http://links.jstor.org.ezproxy.lib.uconn.edu/sici?sici=0091-1798(198405)12:2<283:TSOLRM>2.0.CO;2-O&origin=MSN.
doi: 10.1214/aop/1176993291. |
[7] |
P. J. Forrester,
Lyapunov exponents for products of complex Gaussian random matrices, J. Stat. Phys., 151 (2013), 796-808.
doi: 10.1007/s10955-013-0735-7. |
[8] |
P. J. Forrester, Asymptotics of finite system Lyapunov exponents for some random matrix ensembles, J. Phys. A, 48 (2015), 215205, 17 pp.
doi: 10.1088/1751-8113/48/21/215205. |
[9] |
H. Furstenberg and H. Kesten,
Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.
doi: 10.1214/aoms/1177705909. |
[10] |
H. Furstenberg and Y. Kifer,
Random matrix products and measures on projective spaces, Israel J. Math., 46 (1983), 12-32.
doi: 10.1007/BF02760620. |
[11] |
A. Goswami, Random continued fractions: A Markov chain approach, Econom. Theory, 23 (2004), 85–105, https://doi.org/10.1007/BF02760620.
doi: 10.1007/s00199-002-0333-4. |
[12] |
É. Janvresse, B. Rittaud and T. de la Rue,
How do random Fibonacci sequences grow?, Probab. Theory Related Fields, 142 (2008), 619-648.
doi: 10.1007/s00440-007-0117-7. |
[13] |
É. Janvresse, B. Rittaud and T. de la Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, J. Phys. A, 42 (2009), 085005, 18 pp.
doi: 10.1088/1751-8113/42/8/085005. |
[14] |
É. Janvresse, B. Rittaud and T. de la Rue,
Almost-sure growth rate of generalized random Fibonacci sequences, Ann. Inst. Henri Poincaré Probab. Stat., 46 (2010), 135-158.
doi: 10.1214/09-AIHP312. |
[15] |
V. Kargin,
On the largest Lyapunov exponent for products of Gaussian matrices, J. Stat. Phys., 157 (2014), 70-83.
doi: 10.1007/s10955-014-1077-9. |
[16] |
M. Kieburg and H. Kösters,
Products of random matrices from polynomial ensembles, Ann. Inst. Henri Poincaré Probab. Stat., 55 (2019), 98-126.
doi: 10.1214/17-AIHP877. |
[17] |
R. Lima and M. Rahibe,
Exact Lyapunov exponent for infinite products of random matrices, J. Phys. A, 27 (1994), 3427-3437.
doi: 10.1088/0305-4470/27/10/019. |
[18] |
J. Marklof, Y. Tourigny and L. Wolowski,
Explicit invariant measures for products of random matrices, Trans. Amer. Math. Soc., 360 (2008), 3391-3427.
doi: 10.1090/S0002-9947-08-04316-X. |
[19] |
C. M. Newman, The distribution of Lyapunov exponents: Exact results for random matrices, Comm. Math. Phys., 103 (1986), 121–126, http://projecteuclid.org/euclid.cmp/1104114627.
doi: 10.1007/BF01464284. |
[20] |
Y. Peres, Analytic dependence of Lyapunov exponents on transition probabilities, Lyapunov
Exponents (Oberwolfach, 1990), Lecture Notes in Math., Springer, Berlin, 1486 (1991), 64–
80.
doi: 10.1007/BFb0086658. |
[21] |
Y. Peres, Domains of analytic continuation for the top Lyapunov exponent, Ann. Inst. H. Poincaré Probab. Statist., 28 (1992), 131–148, http://www.numdam.org/item?id=AIHPB_1992__28_1_131_0. |
[22] |
M. Pollicott,
Maximal Lyapunov exponents for random matrix products, Invent. Math., 181 (2010), 209-226.
doi: 10.1007/s00222-010-0246-y. |
[23] |
V. Yu. Protasov and R. M. Jungers,
Lower and upper bounds for the largest Lyapunov exponent of matrices, Linear Algebra Appl., 438 (2013), 4448-4468.
doi: 10.1016/j.laa.2013.01.027. |
[24] |
D. Ruelle,
Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80.
doi: 10.1016/0001-8708(79)90029-X. |
[25] |
D. Viswanath,
Random Fibonacci sequences and the number $1.13198824\ldots$, Math. Comp., 69 (2000), 1131-1155.
doi: 10.1090/S0025-5718-99-01145-X. |
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